Posts Tagged ‘Leo’

About Mystical Number Theory and Pascal’s Triangle

Friday, December 2nd, 2011

The first part of this Blog is about the Triangular numbers, related to the Number 3, the Holy Trinity.

The second part shows that Pascal’s Triangle (called Meru’s Mountain in Mystics), the Binomial Expansion,  contains every Possible Mystical Number Pattern (including the Triangular Numbers) you can Imagine.

Pascal Triangle also shows that our Universe is a combinatorial miracle. It explores every possibility, is always in balance, expands and moves back to the beginning which is and was the Void, the Empty Set, the merge of Every Paradox, that is Possible.

About Mystical Number-Patterns

The Sēpher Yəṣîrâh (Book of Formation or Book of Creation, ספר יצירה) is the oldest book on Jewish Mysticism. The Sefer Yetzirah describes how the universe was created by the “God of Israel” through 32 Wondrous Ways of Wisdom.

The Number 32 is the Sum of the 10 Sephirot and the 22 Letters of the Hebrew Alphabet.

The Sephirot is related to the  Tetraktys of Pythagoras. The Tetraktys embodies the Four main Greek Cyclical (PlatonicMusical Harmonies: the Fourth (4:3), the Ffth (3:2), the Octave (2:1) and the Double Octave (1:4).

1+2+3+4 = 10. 10 is the 4th Triangular Number. The Nth triangular number is the Sum of the numbers 1 -> N. This Sum is equal to 1/2N(N+1).

Between the 10 Sephirot run 22 Channels or Paths which connect them.

The Sephirot are the Points of the Tetraktys. The Hebrew Letters are the Lines between the Points. The Lines of the  Sephirot and the Tetraktys create a Cube (6) at the Top and a Tetrahedron (4) at the Bottom.

The Letters of the Hebrew  Alphabet are divided in the 3 Mother Letters (אמש, the Trinity), the Seven Doubles (The Planets) and the Twelve Simples (the Zodiac).

The 22 letters of the Hebrew Alphabet are a combination of the Trinity, the 7 Planets and 12 Signs of the Zodiac.

When you analyse the Sepher Yeshirah the Cube of Space (the Kaaba) appears out of the Hebrew Alphabet. The Kaaba is related to the Seventh Planet, Saturn.

The 3 Axis of the Cube of Space are the Trinity, the 6 (2×3) Faces of the Cube stand for the Planets with the 7th Saturn, the Son of the Central Sun (3+1 (Center)+3) in the Center and the 12 (4×3) Boundary Lines of the Cube represent the 12 Signs of the Zodiac.

As you can see the Number Three, the Triangle,  plays an important role. It is the First Structure that is Closed in Itself and is therefore Topological related to the Circle. The Circle (and the Triangle) is able to rotate With and Against the Clock. This property is called Spin in Physics.

It is very important to realize that Everything Rotates in our Universe around a Central Object that rotates around another Central Object. The Central Object Gives Time, determinates the Rythm or Harmonics,  of the Rotation Structure.

The Trinity rotates around the Void. The 7 Chakra’s of the Human rotate around the 4th Chakra, the Heart Chakra, ((3+(1)+3)=7).

The Planets rotate around the Sun and the Sun rotates around the Central Black Hole. The arrow of Sagitarius points to this Black Hole.

On a Six Sided Dice the Sum of all the Numbers is Seven (1+6,2+5,3+4). The Sum of the Six Numbers is 3 X 7 = 21. If we add the Center (Saturn) the Number 22 appears.

22/7 is a good approximation of the number π. π relates the Square (and the Cube) to the Circle.

The Cube of Space symbolizes  the Playing Board of the Game of Life. On the Playing Board we have a Free Choice to move into the many Paths that are available. Every Path has its own Probability and this Probability can be calculated. If we don’t know what to do we could throw a Dice.

The Cube of Space contains the same six lines that exist in the I Ching. Four of the lines are of equal length, the other two, the diagonals, are longer. For this reason symmetry cannot be statically produced and the Dance (of Shiva) results.

The Circle represents the Cycles of Time of the Matrix of the Demiurg. Behind all the Probabilities of all the Possible Paths lies a Hidden Order.

A Hexagram, represented by the Star of David,  is a Two-Dimensional (Orthographic) projection of a Cube. A Symmetric Projection of the Cube creates a Cross.

A Hexagram is a Two Dimensional Cube

One of the many meanings of the first word in the Bible “Bereshit“,  is “They (Elohim) created Six” which means that in Six Stages of  the Time Cycle the Cube of Space (or the Hexagram) was populated. On the Seventh Day the Center was filled.

The book of Genesis does not describe the creation of the Trinity (They, Elohim, 1+2+3, 1x2x3) itself. This stage was later covered in the Zohar.

In my blog “About the Sum of Things” it is shown that Six Stages are part of an Expansion Pattern governed by the Powers of Two. After 2**6 (64) Expansions (or Compressions) the Same Fractal Pattern repeats itself on a higher level.

64 is the Number of the I Tjing and the Game of Chess. The number 32 of the Sepher Yeshirah is 64/2 and is a Contraction of the I Tjing.

The I Tjing is a contraction of the oldest Divination System in the Word called FA. FA is still used all over the world by the followers of the oldest wisdom-system created by the YOrubA in Africa. The Yoruba lived at the place where the ancient Paradise was situated.

Star of David in The Israeli Art Genesis-2

The Fourth Day (Sun (4), Moon (5))

About the Triangular Numbers

The Tetraktys contains the Numbers 1, 3, 6 and 10. These numbers are called Triangular Numbers.

The number 21 is also a Triangular Number because it is the Sum of  the Sixth Level of the Tetraktys,  the Numbers 1 to 6.

The Fifth Level of the Tetractys is related to the Number 15 (1+2+3+4+5). This number connects the Tetractys and the Sephirot to the 3×3 Lo Shu Magic Square also called the Seal of Saturn.

The nth Triangle number T(n) is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n.  T(n)=n(n+1)/2.

The Triangular Numbers contain the Perfect Numbers. A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Six (1+2+3=1x2x3) is the first Perfect Number and 28 (1+2+4+7+14) is the next.

The Sum of two Triangular Numbers is a Square

The Sum of two adjacent Triangular Numbers T(n) +  T(n+1) is a Square Number because Two Triangels can be combined in a Square. 1+3=2**2 and 3+6=3**2.

There are many relationships between the Triangular Numbers. These relationships were the focus of the research of the Mystical Group of the Mathematikoi of Pythagoras.

6 (Bereshit, the Cube, the Hexagram) + the 22 Letters of the Hebrew Alphabet = 28, the Next Perfect Number (1+2+3+4+5+6+7).

28 is like the numbers 6 and 15 also a Hexagonal Number. As you can see in the picture below 28 is the fourth Hexagonal Number. As we have seen before a Hexagon is a Projection of a Cube so 28 represents a Cube in a Cube in a Cube. A Cube in a Cube is called a Tessarect or a HyperCube.

28 is a Hexagonal Number

The Number 15 is a Cube in a Cube called a Tessarect or a HyperCube

The first sentence in Genesis (“In the beginning Elohim created Heaven and Eearth“) contains 7 words and 28 letters. This indicates that the Creation Process was already in the 7th stage of the Tetraktys and in its 2nd Fractal Expansion,  the Birth of the Material Universe.

The sum of the entire verse is the 73rd Triangular Number. The prime Numbers 37 and 73 are geometrically related. They form the third and the fourth term in the sequence of Star Numbers (1, 13, 37, 73, 121).

Hexagon/Star pairs are closely related to Triangular numbers. Their product is always a Triangle, and they can be symmetrically generated from a Pair of Triangles.

Star Numbers are a Combination of Two Triangular Numbers

The Square is a combination of two Triangels. It is represented by the Of Star of David, the Symbol of the Heart Chakra.

The Symbol of the Heart Chakra contains Two Triangles.

About Pascal’s Triangle

When a number represents a Geometric Structure it is called a Figurative Number.

Every possible figurative number is generated by the Triangle of Pascal.

The Fractal Sierpinsky Triangle is the Triangle of Pascal Modulo 2.

The Triangle of Pascal was known long before Pascal (re)discovered it.

It was known in Ancient India as the Meru Prastara and in China as  the Yang Hui.  Meru Prastara relates the triangel to a Mystical Mountain called Mount Meru. Mount Meru is also implemented in the Sri Yantra.

The Triangle shows the Coefficients of the Function F(X,Y))= (X+Y)**n. If n=0 F(X,Y)=1 and if n=1 F(X,Y)=X+Y so the Coeffcients are (1,1).

Pascals Triangle is a 2-Dimensional System based on the Polynomal (X+Y)**N. It is always possible to generalize this structure to Higher Dimensional Levels. 3 Variables ((X+Y+X)**N) generate The Pascal Pyramid and n variables (X+Y+Z+….)**N  generate The Pascal Simplex.

The rows of the Pascal’s Triangle add up to the power of 2 of the row. So the sum of row 0 is 2**0 and  the sum of row 1 is 2**1 =2.

The Sum of the  rows of the higher n-dimensional versions of the Triangle is n**N where n is the Amount of Variables and N the level of expansion. So the Sum of Pascal’s Pyramid (3 variables X,Y,Z) is 3**N.

Triangle of Pascal

The most interesting property of the Triangle is visible in the Diagonals.

The First Diagonal contains only 1′s. The Ones represent Unique Objects. They are the Points in the Tetraktys.

The Second Diagonal contains the natural numbers. These Numbers are used to Count Objects that are The Same. The Natural Numbers are the Lines that connect the Points. The Natural Numbers are the Sum of the previous Ones.

The Third Diagonal contains the triangular numbers. The Triangular Numbers are the Sum of the previous Natural Numbers.

This pattern repeats itself all the time.

File:Yanghui triangle.gif

The Yang Hui is an ancient Chinese version of the Triangle of Pascal. This Triangle contains Nine (3x3) Levels.

The Fourth Diagonal contains the tetrahedral numbers (Pyramid Numbers) and the Fifth Diagonal, the pentatope numbers.

Fermat stated that Every Positive Integer is a Sum of at most three Triangular numbers, four Square numbers, five Pentagonal numbers, and n n-polygonal numbers.

The Tetrahedron with basic length 4 (summing up to 20) can be looked at as the 3-Dimensional analogue of the Tetraktys.

File:Pyramid of 35 spheres animation.gif

A Tetrahedral Number represents a 3D-version of the Tetraktys.

The Diagonals of the Triangle of Pascal contain every Possible 2-Dimensional Figurative Number (and Structure).

These Numbers are Projections of Higher Dimensional Numbers and Higher Dimensional Structures.

The Higher Dimensional Versions of the Triangle (the Pascal Pyramid, The Pascal Simplex) contain these structures.

The Rows of the Triangle Sum to the Powers of Two (2 Dimensions). These Powers control the Levels of Expansion.

Every 7th step the Fractal Pattern of the Triangle repeats itself on a higher Level.

The Figurative Numbers are the Geometric Shapes that are created by the Lines of the Natural Numbers that are connecting the Points of the One.

Pascal’s Triangle also contains the numbers of the Fibonacci Sequence (“The Golden Spiral“).

When we take the Modulo 9 (the Digital Root of Pythagoras) of the Numbers of Fibonacci a repeating patterns of 24 steps shows itself that can be represented by a Star Tetrahedron or Stella Octangula. The Star Tetrahedron is a Three Dimensional Star of David.

the Fibonacci Numbers as a Cube.

The StarTetrahedron, shows the Pattern behind the Sequence of Fibonacci.

Every Figurative Number N is the Sum of the Figurative Numbers N-1.  Every Geometric Shape is a combination of all the Previous Geometric Shapes.

This means that Every Geometric Shape is in the end The Sum of the Sum of the Sum of  …. Triangels, Trinities (Elohim) or Triangular Numbers and therefore an Extension of the Tetractys of Pythagoras.

The Expansion of the Whole is a (Fractal)  Combination of Combinations.

The Triangle of Pascal is related to the so called Binomial Theorem which is used in Combinatorics and Probability Theory to describe the Amount of Combinations of a Set of  Objects.

The rows of the Triangle of Pascal also shows the Bell Shaped Pattern of the Normal Distribution.

The Probability Distribution of the Triangle of Pascal converges to the Normal Distribution because of the Central Limit Theorem. Every Row has a Mean of N/2 and a Variance of (N**1/2)/2 which means that with every new row the Mean and the Variance become Bigger and Bigger.

The Triangle of Pascal and therefore the Figurative Numbers describe Everything that is Possible but every Expansion of the Triangle is less Likely to Occur.

The Triangle of Pascal Modulo 3

The Triangle of Pascal Mod 3 represents the Tetraktys in the Tetraktys in the .....

Because of the Fractal Expansion/Contraction Pattern The Cube of  Space, related to the Element Earth,  explains Everything there is to Know on Our Level of Existence, Mother Earth.

The interesting part of the Figurative Numbers is that they representent Visual Patterns with which we can Reason.

We don’t need complex formulas because we can See what is Possible.

The interesting part of the Triangle of Pascal is that we can See that the Complex Figurative Structures are created out of a very Simple Structure, the Triangle.

If we want to understand our Reality we have to begin with looking at the Beginning and not start somewhere in the Middle.

If we look at the Fractal Expansion Pattern of the Triangle we See that Every new Stage is an Expansion Out of the Middle.

The Expansion of the Human, the Next Step in our Evolution,  is therefore an Expansion Out of the Heart, the Balance of Father Sky and Mother Earth.

Life is not only about Me and the Other.

Life is also about the Relationship between Me and the Other.

If we don’t Collaborate the Next stage in our Evolution will never happen.


The Content of the Sepher Yesirah

About the Sepher Yesirah

About the Cube of Space

About the Tetractys

About the Cube of Space and Psychology

About the Sepher Yesirah and the I Tjing

A correspondence table of the Cube of Space

About Bereshit

About Genesis

About Patterns in the Bible

About Saturn

About the Trinity

About the Sri Yantra and Plato

About the Lo Shu and the I Tjing

All kind of strange relationships between Triangular Numbers

A website about Mystical Number Theory

About the Figurative Numbers

About Combining the Combinations

About the Golden Spiral and Plato

About the Logic of Creation

About Pascal’s Triangle and the Normal Distribution

A complete course in elementary Number Theory

About the Psychology of the Cube of Space

About the Tetraktys and the Zodiac

About the Process Theory of Paul Young

About the Theory of Dewey B. Larsson

Mysteries of the Equilateral Triangle

About Visual Patterns in Number Theory

About Pascal’s Triangle and Cell Division

About Celtic Christianity

Wednesday, November 12th, 2008

TetramorphThe tetramorphs are four angelic beings, drawn from Babylonian symbolism in a vision of the Hebrew Prophet Ezekiel:

As for the likeness of their faces, they four had the face of a man, and the face of a lion, on the right side: and they four had the face of an ox on the left side; they four also had the face of an eagle

The beasts are later described in the Revelation of John: “And the first beast was like a lion, and the second beast like a calf, and the third beast had a face as a man, and the fourth beast was like a flying eagle.”

The four Babylonian symbols represent the four fixed signs, the Cross of the Zodiac, Leo, Taurus (Bull), Scorpio (Eagle) and Aquarius (Human).

They are related to the four ancient elements of Air, Water, Earth, and Fire, the four evangelists (Mark/Lion, Luke/Bull, John/Eagle, Matthew/Human), the four Seasons and the four possible Views on the world (Rules (Lion), Senses (Bull), Emotions (Human) and the Imagination (Aquarius) of Will McWhinney.

zodiacThe Twelve parts of the Zodiac are the twelve (4×3) possible combinations of the Four Tetramorph.

The tetramorph themselves are combinations of the same aspect (the Bull = SensesXSenses).

Will McWhinney calls the twelve combinations “Games” or “Conflicts” because it is impossible to Unite the Four Tetramorphs.

They represent a Tension or a Force. We have to learn to Balance the Four Forces so they are able to Unite in the Center, The One (the Thirteen).

The Zodiac denotes an annual cycle. The constellations also move in a cycle, called the Precession, with a duration of 25,800 years. The Precession is moving retrograde (in the opposite direction) of the movement of the Sun in the yearly Zodiac.

The precession is an indicator of the cultural or religious state of our World. During the phase of the Pisces many Fish- and Water related symbols were used.

It is very interesting to see that the Roman Church is based on the worldview of a very old precessional symbol, the symbol of St Peter and St Mark, the Lion. In the Roman Church the Sins (the Senses) of the Bull have to obey the Moral Rules of the Lion.

Starting about 200 BC, Pisces, The Fish, became the ruler of the cycle of precession. Love thou Neighbor, became the object of Early Christianity. The Roman Church, an extension of the Empire of Rome, was never able to implement the Social Vision of the Fish.

It remained a hierarchical controlled structure with the Pater (Peter), the highest rank in the hierarchy of Mithras (The Killer of the Bull), at the top.

We are now moving from the State of Water into the State of Air (Imagination, Creation). The Social Structures of the Church will be replaced by Connected Individuals who want to Share and Express a Personal Mystic Experience.

The view of John the Evangelist, the view of the Mystic and the Poet, is hidden in the Inner World, the Imagination, of the Human. According to the Mystics the Outer World is an Expression of the Creative Power of this Inner World.

John the Evangelist was a disciple of John the Baptist before he became a disciple of Jesus. John the Baptist used Water when he baptized Jesus. He initiated John the Evangelist and Jezus into the Gnostic Mysteries. Jezus relates John the Baptist to the prophet Elija (Elias). (“there has not been raised of those born of women greater than John the Baptist. … if you are willing to accept it, he is Elias … the one about to come.” (Mt 11:9-14)).

John the BaptistThe symbol of John the Baptist was not the Eagle but its antagonist, the Dove. When we look at the symbolism of the Dove we are able to detect who John the Baptist really was. John the Baptist represents Elija, Vishnu, Oannes, Enki and Ea, the Creator of our Universe, the Lord of the Waves. The Dove is represented in the Zodiac by the Pleiades, the Center of the Cross of the Universe.

In the next part I will give you a very brief introduction into the Believe System of the Mystics.

It is very important to understand that the Mystics believe that the Creator of our Universe is not the creator of the Whole, the creation itself. The process of Creation is an Act of the Imagination of the One, a formless Entity also called The Light or the First Logos, the Word. The One has delegated the process of creation to other creators who are an emanation of the One.

In the beginning was the Word (Logos), and the Word was with God, and the Word was God“.

A creation of a creator is able to create his own reality. One of the creations is Earth and the Humans on Earth are also able to create their own reality with their Imagination which is an expression of the Imagination of the One.

The One splits itself in many parts. The first creation, the Second Logos, is the Two, The Cosmic Father and Mother (Light, Darkness, Yin, Yang). The One, The Cosmic Father and the Mother are the Trinity represented by the Letters I (I am the One), A (The Beginning) and O (the End). (Revelation 1: 8: I am, the Alpha and the Omega). IOA comes back in the name Ioa(ness),the Fish-man, John and the IOROBA of Africa.

The Human is the last step in the creation. He is a five-fold creature, a combination of the Four Forces of the tetramorphs and the Light of the One.

Our Universe is one of the creations of a creator called the Demiurge (Ea, Enki). The Human Soul is in exile in the creation of this Demiurge. There are many universes and Earth is part of one of these universes. Earth is part of the (Dark) Material Universe.

The Soul, the Sparkle of the Light, will eventually return to the One. It is helped to return to the One by other Souls (Angels, Archangels) to reach ‘the Worlds of Light’. One day the world will be saved when Oannes, Enki, Vishnu, Jesus comes back.

Our Reality is governed by a Cyclic Process. This process is created by the Demiurg to keep the Souls in the Black Material Universe. Planets and stars influence fate and are also places of detention after death.

The Book of Destiny, the Book of Revelations, created by the Demiurg and written in the language of ancient Astrology, describes the Cycle. The Souls have to Jump out of the Cycle, through the Hole in the Universe, to reach the World of Light of the One. The Hole in the Universe is in the Centre of the Rose, The Heart.

Mysteries (Rituals, Sacraments) purify the soul, to ensure her rebirth into a spiritual body, and her ascent from the world of matter. The Knowledge of the Mysteries is hidden. The Hidden Mysteries are revealed by the Light, the First Logos, at the time when the Seeker has stopped looking for the Answer to all his Questions. At this moment of complete silence The Light Leads the Way.

John the Evangelist and his mystic woldview can be found in many old and almost extinct religions: Manicians, Coptic, Druze , Alawites, Essenes, Mandaeans, Chaldeans, Valentinians ,Cathars, Sufists and Armenians.

IOnA is a small island in the Inner Hebrides of Scotland (Scota) that has an important place in the history of Christianity. Iona formerly named Innis nan Druidhneach (the Island of the Druids) was a primary seat of learning for the pre-Christian Celtic Druid Magi until St. Columba, the Dove, was exiled from his native Ireland (Tara).

st_columbaSt. Columba founded a monastery on Iona with 12 companions. Iona’s fame as a place of learning and Christian mission spread throughout Europe. Iona became a holy island where several kings of Scotland, Ireland and Norway came to be buried.

St. Columba later came back to Kells, close to Tara, and created another Monastery. One of the most important relics that was taken from Iona to Kells is the Book of Kells.

The most important mystical place in Ireland is called Tara. It is, according to legend, the place where the followers of Ioaness, guarded by the prophet Jeremiah (586 BC), accompanied by his scribe Baruch, and the daughters of Zedekiah, the last king of Judah, went in exile when Jerusalem was taken by the Babylonians under Nebuchadnezzar.

One of the daughters (Tea Tepi, Scota) marries Eochaidh the high king of Ireland. Jeremiah brings many important relics to Ireland including Jacob’s Pillow, the cornerstone of the temple of Salomon.

The religion that came out of Kells and Iona was very different from the religion that came out of Rome. It was the religion of the Celtic Druids AND the followers of John the Baptist called Celtic Christianity.

Celtic Christianity was completely destroyed by the English Rulers of Ireland and Scotland. Just like the Roman Church destroyed the Cathars and Islam destroyed the Druze. Celtic Christianity went underground and came back as The Kabbalists, the Alchemists, the Rosicrucian’s and the Free Masons.

We are now entering a new phase in the Zodiac called Aquarius (Imagination). The view of John the Evangelist will become the Spirit of the Time. It is not necessary to create a New System of Rules, a Religion. The Dove is silently waiting in our Inner World to move Outside to bring Peace to the World. The Rose of the Heart will slowly open without any outside pressure.


About Celtic Christianity

About Celtic Mythology

About Celtic History

About the IORABA